ProemOne of the many intriguing observations in Miranda Lund's book Sacred Geometry is that there are only three regular tilings of a plane. That is, triangles, squares, and hexagons are the only three regular polygons that can completely fill a plane if repeated over and over. There are other ways of filling a plane with combinations of two or more different types of regular polygons, known as semiregular and demiregular tilings. But these semi and demiregular tilings involve only triangles, squares, hexagons, octagons, or dodecagons: that is, 3, 4, 6, 8, and 12 sided figures. What became of the 5, 7, 9, 10, and 11 sided figures? Assuming that a desperate soul were to try to fill a plane with these figures as regularly as possible, what is the best they could do? It turns out there are a good many visually interesting arrangements of polygons across a plane that are structurally interesting, without being a regular, demiregular, or semiregular tiling. Here's an example that is based on pentagons: Some of the interesting properties of this arrangement are:
I don't know of a mathematical term for tilings with these types of properties, but I have discovered (or possibly rediscovered) many of them by placing regular polygons point to point. For a time, it became an obsession of sorts to enumerate these various patterns. Polygon Madness is the record of my experiments with such figures. 



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